Problem: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{-9r^2 - 144r - 576}{-10r^3 - 100r^2 - 160r}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {-9(r^2 + 16r + 64)} {-10r(r^2 + 10r + 16)} $ $ z = \dfrac{9}{10r} \cdot \dfrac{r^2 + 16r + 64}{r^2 + 10r + 16} $ Next factor the numerator and denominator. $ z = \dfrac{9}{10r} \cdot \dfrac{(r + 8)(r + 8)}{(r + 8)(r + 2)}$ Assuming $r \neq -8$ , we can cancel the $r + 8$ $ z = \dfrac{9}{10r} \cdot \dfrac{r + 8}{r + 2}$ Therefore: $ z = \dfrac{ 9(r + 8)}{ 10r(r + 2)}$, $r \neq -8$